The technique is based on detection of the beating between the resonant carrier and a resonant higher order mode.
This means that the beat signal is cancelled out if the transmitted beam is integrated over the entire beam.
Thus, you may want to introduce intentional clipping (or cutting a half of the beam with a razor blade).
Ref: LIGO DCC G080467: Precise Measurements on Longitudinal and Transverse Mode Spacings of an Optical Cavity using an Auxiliary Laser
I am quite curious on the measurement as I am going to do the same measurement for the aLIGO OMC.
I am interested in seeing the statistical evaluation on the precision of the measurement.